arXiv subjects

Marco Rampazzo

Publications and source records attributed to Marco Rampazzo.

11 recordsLinked to original sources

arXiv2026

Categorical reconstruction of del Pezzo surfaces: Hochschild--Serre algebras and spinor modifications

We prove that, for every smooth complex del Pezzo surface of degree at most four, the enhanced right orthogonal to the structure sheaf determines the surface up to isomorphism. In degrees one, two, and three, we recover the anticanonical equation from intrinsic pieces of the Hochschild-Serre algebra via graded matrix factorizations; in degree four, the relevant Serre diagonal recovers the orbifold canonical ring associated with the pencil of quadrics. We also give an alternative proof in degrees one and two using the Bertini and Geiser involutions, equivariant topological K-theory, and classical Torelli. Then, we study the Clifford component associated with a conic bundle structure over the projective line. In every degree at most four, we construct an abstract spinor bundle whose spinor modification produces another, generally non-isomorphic, del Pezzo surface. We show via Kuznetsov's modification theorem that the relevant base-linear equivalences are precisely those induced by spinor modifications.

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arXiv2026

Categorical resolutions and birational geometry of nodal Gushel-Mukai varieties

An ordinary Gushel-Mukai variety with a single isolated node is the intersection of the Grassmannian $G(2, 5)$ with a nodal quadric and a linear space. We consider such intersections in dimension three, four and five. We describe a flop between the blowup of such a variety and a quadric fibration over $\mathbb{P}^2$: at the level of derived categories, this flop establishes an equivalence between the categorical resolution of the Kuznetsov component of the Gushel-Mukai variety and the derived category of modules on the even part of the Clifford algebra of the quadric fibration. As a first application, we extend a result of Kuznetsov and Perry to the nodal case, and we describe a subfamily of rational, nodal Gushel-Mukai fourfolds whose Kuznetsov components admit a categorical resolution of singularities by an actual K3 surface of degree two without a Brauer twist. This produces evidence for a version of Kuznetsov's rationality conjecture. We also describe the relation with Verra threefolds and fourfolds at the birational and categorical level. In particular, in the three-dimensional case, we investigate alternative birational models by hyperbolic equivalence and by Kuznetsov's spinor modifications. We show that the categorical resolution of the Kuznetsov component of a one-nodal Gushel-Mukai threefold determines its birational class, and we explicitly construct another birational model, which is again a conic fibration over $\mathbb{P}^2$ branched over a sextic, with the same Kuznetsov component up to autoequivalences.

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arXiv2025

Window categories for a simple $9$-fold flop of Grassmannian type

The local simple $9$-fold flop of Grassmannian type is a birational transformation between total spaces of vector bundles on the Grassmannians $\mathrm{Gr}(2, 5)$ and $\mathrm{Gr}(3, 5)$. We produce four different derived equivalences which commute with the pushforward functors for the flopping contractions. These equivalences are realized by identifying four different window categories inside the derived category of coherent sheaves on an Artin stack. As an application, our approach provides a new proof of derived equivalence for a pair of non-birational Calabi-Yau threefolds realized as zero loci of sections of homogeneous vector bundles in Grassmannians.

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arXiv2024

Derived equivalence for the simple flop of type $D_5$

We prove that every simple flop of type $D_5$, i.e., resolved by blowups with exceptional divisor isomorphic to a generalized Grassmann bundle with fiber $OG(4, 10)$, induces a derived equivalence. This provides new evidence for the DK conjecture of Bondal--Orlov and Kawamata. The proof is based on a sequence of mutations of exceptional objects: we use the same argument to prove derived equivalence for some pairs of non-birational Calabi--Yau fivefolds in $OG(5, 10)$, related to Manivel's double--spinor Calabi--Yau varieties. We extend the construction to prove the derived equivalence of Calabi--Yau fibrations, which are described as zero loci in some generalized Grassmann bundles.

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arXiv2024

Fano fibrations and DK conjecture for relative Grassmann flips

Given a vector bundle $\mathcal E$ on a smooth projective variety $B$, the flag bundle $\mathcal F l(1,2,\mathcal E)$ admits two projective bundle structures over the Grassmann bundles $\mathcal G r(1, \mathcal E)$ and $G r(2, \mathcal E)$. The data of a general section of a suitably defined line bundle on $\mathcal F l(1,2,\mathcal E)$ defines two varieties: a cover $X_1$ of $B$ and a fibration $X_2$ on $B$ with general fiber isomorphic to a smooth Fano variety. We construct a semiorthogonal decomposition of the derived category of $X_2$ which consists of a list of exceptional objects and a subcategory equivalent to the derived category of $X_1$. As a byproduct, we obtain a new full exceptional collection for the Fano fourfold of degree $12$ and genus $7$. Any birational map of smooth projective varieties which is resolved by blowups with exceptional divisor $\mathcal F l(1, 2, \mathcal E)$ is an instance of a so-called Grassmann flip: we prove that the DK conjecture of Bondal-Orlov and Kawamata holds for such flips. This generalizes a previous result of Leung and Xie to a relative setting.

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arXiv2023

Fullness of the Kuznetsov-Polishchuk exceptional collection for the spinor tenfold

Kuznetsov and Polishchuk provided a general algorithm to construct exceptional collections of maximal length for homogeneous varieties of type A,B,C,D. We consider the case of the spinor tenfold and we prove that the corresponding collection is full, i.e. it generates the whole derived category of coherent sheaves. As a step of the proof, we construct some resolutions of homogeneous vector bundles which might be of independent interest.

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arXiv2022

New counterexamples to the birational Torelli theorem for Calabi--Yau manifolds

We produce counterexamples to the birational Torelli theorem for Calabi-Yau manifolds in arbitrarily high dimension: this is done by exhibiting a series of non birational pairs of Calabi-Yau $(n^2-1)$-folds which, for $n \geq 2$ even, admit an isometry between their middle cohomologies. These varieties also satisfy an $\mathbb L$-equivalence relation in the Grothendieck ring of varieties, i.e. the difference of their classes annihilates a power of the class of the affine line. We state this last property for a broader class of Calabi-Yau pairs, namely all those which are realized as pushforwards of a general $(1,1)$-section on a homogeneous roof in the sense of Kanemitsu, along its two extremal contractions.

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arXiv2022

The generalized roof F(1,2,n): Hodge structures and derived categories

We consider generalized homogeneous roofs, i.e. quotients of simply connected, semisimple Lie groups by a parabolic subgroup, which admit two projective bundle structures. Given a general hyperplane section on such a variety, we consider the zero loci of its pushforwards along the projective bundle structures and we discuss their properties at the level of Hodge structures. In the case of the flag variety $F(1,2,n)$ with its projections to $\mathbb{P}^{n-1}$ and $G(2, n)$, we construct a derived embedding of the relevant zero loci by methods based on the study of $B$-brane categories in the context of a gauged linear sigma model.

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arXiv2021

Mukai duality via roofs of projective bundles

We investigate a construction providing pairs of Calabi-Yau varieties described as zero loci of pushforwards of a hyperplane section on a roof as described by Kanemitsu. We discuss the implications of such construction at the level of Hodge equivalence, derived equivalence and $\mathbb L$-equivalence. For the case of $K3$ surfaces, we provide alternative interpretations for the Fourier-Mukai duality in the family of $K3$ surfaces of degree 12 of Mukai. In all these constructions the derived equivalence lifts to an equivalence of matrix factorizations categories.

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arXiv2021

Calabi-Yau fibrations, simple K-equivalence and mutations

A homogeneous roof is a rational homogeneous variety of Picard rank 2 and index $r$ equipped with two different $\mathbb P^{r-1}$-bundle structures. We consider bundles of homogeneous roofs over a smooth projective variety, formulating a relative version of the duality of Calabi--Yau pairs associated to roofs of projective bundles. We discuss how derived equivalence of such pairs can lift to Calabi--Yau fibrations, extending a result of Bridgeland and Maciocia to higher-dimensional cases. We formulate an approach to prove that the $DK$-conjecture holds for a class of simple $K$-equivalent maps arising from bundles of roofs. As an example, we propose a pair of eight-dimensional Calabi--Yau varieties fibered in dual Calabi--Yau threefolds, related by a GLSM phase transition, and we prove derived equivalence with the methods above.

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arXiv2019

Torelli problem for Calabi-Yau threefolds with GLSM description

We construct a gauged linear sigma model with two non-birational Kälher phases which we prove to be derived equivalent, $\mathbb{L}$-equivalent, deformation equivalent and Hodge equivalent. This provides a new counterexample to the birational Torelli problem which admits a simple GLSM interpretation.

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